4.2 Lensing in conformally stationary spacetimes

Conformally stationary spacetimes are models for gravitational fields that are time-independent up to an
overall conformal factor. (The time-dependence of the conformal factor is important, e.g., if cosmic
expansion is to be taken into account.) This is a reasonable model assumption for many, though not all,
lensing situations of interest. It allows describing light rays in a 3-dimensional (spatial) formalism that will
be outlined in this section. The class of conformally stationary spacetimes includes spherically symmetric
and static spacetimes (see Sections 4.3) and axisymmetric stationary spacetimes (see Section 4.4). Also,
conformally flat spacetimes (see Section 4.1) are conformally stationary, at least locally. A physically
relevant example where the conformal-stationarity assumption is not satisfied is lensing by a gravitational
wave (see Section 5.11).

By definition, a spacetime is conformally stationary if it admits a timelike conformal Killing vector field
. If is complete and if there are no closed timelike curves, the spacetime must be a product,
with a (Hausdorff and paracompact) 3-manifold and parallel to the
-lines [147]. If we denote the projection from to by and choose local coordinates
on , the metric takes the form

with . The conformal factor does not affect the lightlike geodesics apart from their
parametrization. So the paths of light rays are completely determined by the metric
and the one-form which live on . The metric must be positive definite to give a
spacetime metric of Lorentzian signature. We call the redshift potential, the Fermat metric and
the Fermat one-form. The motivation for these names will become clear from the discussion
below.

If , where is a function of , we can change the time coordinate according
to , thereby transforming to zero, i.e., making the surfaces
orthogonal to the -lines. This is the conformally static case. Also, Equation (61) includes the
stationary case ( independent of ) and the static case ( and independent of
).

In Section 2.9 we have discussed Kovner’s version of Fermat’s principle which characterizes the lightlike
geodesics between a point (observation event) and a timelike curve (worldline of light source) . In
a conformally stationary spacetime we may specialize to the case that is an integral curve of the
conformal Killing vector field, parametrized by the “conformal time” coordinate (in the past-pointing
sense, to be in agreement with Section 2.9). Without loss of generality, we may assume that the observation
event takes place at . Then for each trial path (past-oriented lightlike curve)
from to the arrival time is equal to the travel time in terms of the time function
. By Equation (61) this puts the arrival time functional into the following coordinate form

where is any parameter along the trial path, ranging over an interval that depends on the
individual curve. The right-hand side of Equation (62) is a functional for curves in with fixed
end-points. The projections to of light rays are the stationary points of this functional. In general, the
right-hand side of Equation (62) is the length functional of a Finsler metric. In the conformally static
case , the integral over is the same for all trial paths, so we are left
with the length functional of the Fermat metric . In this case the light rays, if projected to
, are the geodesics of . Note that the travel time functional (62) is invariant under
reparametrization; in the terminology of classical mechanics, it is a special case of Maupertuis’principle. It is often convenient to switch to a parametrization-dependent variational principle
which, in the terminology of classical mechanics, is called Hamilton’s principle. The Maupertuis
principle with action functional (62) corresponds to Hamilton’s principle with a Lagrangian

where are the Christoffel symbols of the Fermat metric . The solutions admit the constant of
motion

which can be chosen equal to 1 for each ray, such that gives the -arclength. By Equation (62),
the latter gives the travel time if . According to Equation (64), the Fermat two-form

exerts a kind of Coriolis force on the light rays. This force has the same mathematical structure as the
Lorentz force in a magnetostatic field. In this analogy, corresponds to the magnetic (vector)
potential. In other words, light rays in a conformally stationary spacetime behave like charged
particles, with fixed charge-to-mass ratio, in a magnetostatic field on a Riemannian manifold
.

Fermat’s principle in static spacetimes dates back to Weyl [347] (cf. [207, 318]). The stationary case
was treated by Pham Mau Quan [276], who even took an isotropic medium into account, and later, in a
more elegant presentation, by Brill [42]. These versions of Fermat’s principle are discussed in several
text-books on general relativity (see, e.g., [226, 115, 311] for the static and [199] for the stationary
case). A detailed discussion of the conformally stationary case can be found in [265]. Fermat’s
principle in conformally stationary spacetimes was used as the starting point for deriving the
lens equation of the quasi-Newtonian apporoximation formalism by Schneider [296] (cf. [298]).
As an alternative to the name “Fermat metric” (used, e.g., in [115, 311, 265]), the names
“optical metric” (see, e.g., [140, 79]) and “optical reference geometry” (see, e.g., [4]) are also
used.

In the conformally static case, one can apply the standard Morse theory for Riemannian geodesics to the
Fermat metric to get results on the number of -geodesics joining two points in space. This
immediately gives results on the number of lightlike geodesics joining a point in spacetime to an integral
curve of . Completeness of the Fermat metric corresponds to global hyperbolicity of the spacetime
metric. The relevant techniques, and their generalization to (conformally) stationary spacetimes, are
detailed in a book by Masiello [219]. (Note that, in contrast to standard terminology, Masiello’s definition
of a stationary spacetime includes the assumption that the hypersurfaces are
spacelike.) The resulting Morse theory is a special case of the Morse theory for Fermat’s principle in
globally hyperbolic spacetimes (see Section 3.3). In addition to Morse theory, other standard
methods from Riemannian geometry have been applied to the Fermat metric, e.g., convexity
techniques [138, 139].

If the metric (61) is conformally static, , and if the Fermat metric is conformal to
the Euclidean metric, , the arrival time functional (62) can be written as

where is Euclidean arclength. Hence, Fermat’s principle reduces to its standard optics form for an
isotropic medium with index of refraction on Euclidean space. As a consequence, light propagation in a
spacetime with the assumed properties can be mimicked by a medium with an appropriately chosen index of
refraction. This remark applies, e.g., to spherically symmetric and static spacetimes (see Section 4.3) and,
in particular, to the Schwarzschild spacetime (see Section 5.1). The analogy with ordinary optics in media
has been used for constructing, in the laboratory, analogue models for light propagation in
general-relativistic spacetimes (see [243]).

Extremizing the functional (67) is formally analogous to Maupertuis’ principle for a particle in a scalar
potential on flat space, which is discussed in any book on classical mechanics. Dropping the assumption that
the Fermat one-form is a differential, but still requiring the Fermat metric to be conformal to the Euclidean
metric, corresponds to introducing an additional vector potential. This form of the optical-mechanical
analogy, for light rays in stationary spacetimes whose Fermat metric is conformal to the Euclidean metric, is
discussed, e.g., in [7].

The conformal factor in Equation (61) does not affect the paths of light rays. However, it does
affect redshifts and distance measures (recall Section 2.4). If is of the form (61), for every lightlike
geodesic the quantity is a constant of motion. This leads to a particularly simple form of the
general redshift formula (36). We consider an arbitrary lightlike geodesic in terms of its
coordinate representation . If both observer and emitter are at rest in the
sense that their 4-velocities and are parallel to , Equation (36) can be rewritten as

This justifies calling the redshift potential. It is shown in [150] that there is a redshift potential for a
congruence of timelike curves in a spacetime if and only if the timelike curves are the integral curves of a
conformal Killing vector field. The notion of a redshift potential or redshift function is also
discussed in [74]. Note that Equation (68) immediately determines the redshift in conformally
stationary spacetimes for any pair of observer and emitter. If the 4-velocity of the observer or of the
emitter is not parallel to , one just has to add the usual special-relativistic Doppler
factor.

Conformally stationary spacetimes can be characterized by another interesting property. Let be a
timelike vector field in a spacetime and fix three observers whose worldlines are integral curves of .
Then the angle under which two of them are seen by the third one remains constant in the course of time,
for any choice of the observers, if and only if is proportional to a conformal Killing vector field. For a
proof see [150].